Embed Size (px)
Citation preview
Institute for Computational Mathematics Hong Kong Baptist University
ICM Research Report 11-03
High Order Weighted Essentially Non-Oscillation Schemes for
Two-Dimensional Detonation Wave Simulations
Zhen Gao ∗, Wai Sun Don †, Zhiqiu Li ‡
August 1, 2011
Abstract
In this paper, we demonstrate the detailed numerical studies of three classical two dimension-al detonation waves by solving the two dimensional reactive Euler equations with species withthe fifth order WENO-Z finite difference scheme (JCP 227, 2008) with various grid resolutions.To reduce the computational cost and to avoid wave reflection from the artificial computationalboundary of a truncated physical domain, we derive an efficient and easily implemented onedimensional Perfectly Matched Layer (PML) absorbing boundary condition (ABC) for the twodimensional unsteady reactive Euler equation when one of the directions of domain is periodicaland inflow/outflow in the other direction. The numerical comparison among characteristic, freestream, extrapolation and PML boundary conditions are conducted for the detonation wavesimulations. The accuracy and efficiency of four mentioned boundary conditions are verifiedagainst the reference solutions which are obtained from using a large computational domain.Numerical schemes for solving the system of hyperbolic conversation laws with a single-modesinusoidal perturbed ZND analytical solution as initial conditions are presented. Regular rect-angular combustion cell, pockets of unburned gas and bubbles and spikes are generated andresolved in the simulations. It is shown that large amplitude of perturbation wave generatesmore fine scale structures within the detonation waves and the number of cell structures dependson the wave number of sinusoidal perturbation.
KeywordsWeighted Essentially Non-Oscillatory, Detonation, ZND, Perfectly Matched Layer
AMS65P30, 77Axx
∗Correspondence author: School of Mathematical Sciences, Ocean University of China, Qingdao, China. E-Mail:[email protected]
†Department of Mathematics, Hong Kong Baptist University, Hong Kong, China. E-Mail: [email protected]‡Department of Mathematics, Hong Kong Baptist University, Hong Kong, China. E-Mail: [email protected]
1
1 Introduction
Conservative Weighted Essentially Non-Oscillation finite difference schemes (WENO) have beendeveloped in recent years as a class of high order/high resolution method for solutions of hyperbolicconservation laws (PDEs) in the presence of shocks and small scale structures in the solution (fordetails and history of WENO scheme, see [1] and references contained therein). WENO schemesowe their success to the use of a dynamic set of substencils where a nonlinear convex combinationof lower order polynomials adapts either to a higher order polynomial approximation at smoothparts of the solution, or to a lower order polynomial approximation that avoids interpolation acrossdiscontinuities. The upwinding of the spatial discretization provides the necessary dissipation forshock capturing. It is an evolution of the Essentially Non-Oscillatory (ENO) schemes, introducedin [2], which choose only the smoothest substencil, instead of forming a convex combination of allthe substencils available in order to optimize accuracy in the smooth parts of the solution.
The local computational stencils of (2r − 1) order WENO schemes are composed of r overlappingsubstencils of r points, forming a larger stencil with (2r − 1) points. The scheme yields a localrate of convergence that goes from order r at the non-smooth parts of the solution, to order(2r − 1) when the convex combination of local lower order polynomials is applied at smooth partsof the solution. The nonlinear coefficients of WENO’s convex combination, hereafter referred toas nonlinear weights ωk, are based on lower order local smoothness indicators βk, k = 0, . . . , r − 1that measure the sum of the normalized squares of the scaled L2 norms of all derivatives of r localinterpolating polynomials. An essentially zero weight is assigned to those lower order polynomialswhose underlining substencils contain high gradients and/or shocks, aiming at an essentially non-oscillatory solution close to discontinuities. At smooth parts of the solution, the formal orderof accuracy is achieved through the mimicking of the central upwind scheme of maximum order,when all smoothness indicators are about the same size. Hence, the one of the most importantissues for WENO schemes is to design an efficient and accurate nonlinear weights ωk. In [3], thefirst set of nonlinear weights of widespread use has been given. However, it has been shown thatthe nonlinear weights fail to satisfy the necessary and sufficient condition for achieving the formalorder of accuracy even for smooth functions. We call this scheme as the classical WENO scheme(WENO-JS). In [4], a modification of the nonlinear weights was proposed in the form of a mappingon the classical WENO-JS nonlinear weights, leading to corrected nonlinear weights that recoveredthe formal order of accuracy. We call the scheme composed by this mapped set of nonlinearweights as the mapped WENO scheme (WENO-M). In [5], it was shown that the incorporation ofa global optimal order smoothness indicator, hereafter denoted as τ2r−1, into the classical WENO-JSnonlinear weights definition satisfies the necessary and sufficient condition for achieving the formalorder of accuracy. This scheme has been named the WENO-Z scheme (WENO-Z). The mappingprocedure of WENO-M incurs extra expensive computational cost, while the weight modification ofWENO-Z is obtained through a simple and inexpensive linear combination of the already computedlower order local smoothness indicators βk. It has been shown that the new set of nonlinear weightsof WENO-Z provided less dissipation than WENO-JS and yielded comparable resolution of smoothsolution and captured sharp gradients as WENO-M [5].
2
Since the classical theory, namely the ZND detonation model, in detonation waves was pioneeredby Zekdivich [6], von Neumann [7] and Doering [8], the studies of detonation waves have been anactive area of research in both theoretical studies and in numerical simulations due to the practicalimportance. For instance, linear stability analysis [9] presented rigorous results, but it is unable tocapture properties of non-linear dynamics. In the 1970s, experiments [10, 11, 12] demonstrated thatthe detonations observed in many circumstances exhibited complicated unstable wave patterns inreacting gases. Numerical approaches such as the second order Godunov scheme[13, 14], extendedCE/SE method [15], unsplit scheme [16], classical WENO-JS scheme [17, 18] have been implementedto simulate two dimensional detonation waves to investigate the detonation phenomenon in manyphysical applications. Moreover, the grid convergence study for the case of overdrive factor f = 1.6in [19] showed that the WENO schemes converge faster than existing numerical methods such asPPM with front tracking and mesh refinement [20], unsplit scheme [21], Roe’s solver with minmodlimiter [22] and Roe’s solver with superbee limiter [22]. Consequently, in this paper we are interestedin employing the fifth order WENO-Z schemes for the numerical simulations of two dimensionaldetonation waves.
In simulating detonation waves in a semi-infinite physical domain, the computational domain mustbe truncated which necessitates the imposition of numerical boundary condition at the artificialoutflow boundary in order to close the system of equations. Unless a large computational domain isused for simulations so that detonation waves do not reach the artificial boundary before the finaltime is reached, waves reflected from the artificial boundary can be reflected back into the interiorcomputational domain. The reflected waves will interact with the detonation waves and pollutethe developing solution. Berenger, in [23], originally designed a Perfectly Matched Layer (PML)for Maxwell’s equation to truncate an infinity or semi-infinity computational domain. It works as arelatively simple and efficient absorbing boundary condition for the wave problems. In the last fewyears, it was analyzed and developed in linearized and nonlinear Euler equations [24, 25, 26, 27] toabsorb the outgoing waves and decrease the computational cost. In this study, we design an efficientand easily implemented x-direction PML for the unsteady reactive Euler equations to truncate thecomputational domain for two dimensional detonation waves. The additional cost of PML equationis negligible because the original Euler equation is only appended with an extra source term. Theaccuracy and efficiency of designed PML boundary condition are verified by comparing it with thecharacteristic, extrapolation, free stream boundary conditions and computing the pressure erroragainst the reference solutions, which are obtained from a simulation with a large computationaldomain under the same grid resolution.
In this study of detonation wave simulations, we consider three classical examples listed in thefollowing Table I, where γ is the ratio of specific-heat, Ea is the activation-energy parameter, q0 isthe heat-release parameter, f is the overdrive factor.
We perturb the planar ZND wave by perturbing the vertical velocity v in y inside the half-reactionzone in the initial condition with a single-mode sinusoidal perturbation with small and large am-plitude of perturbation. In the case (H2, E20, f1.1), solutions converge to steady state for highresolutions δ10 and δ20 (which denotes the resolution as 20 grid points per half-reaction zone), and
3
Cases studied Abbreviation of cases γ q0 Ea f
Small heat release/Low activation energy (H2, E20, f1.1) 1.2 2 20 1.1
Large heat release/Low activation energy (H50, E10, f1.2) 1.2 50 10 1.2
Large heat release/Large activation energy (H50, E50, f1.2) 1.2 50 50 1.2
Table I: Cases of simulation and corresponding parameters.
the so-called regular rectangular modes are formed. In the more interesting case (H50, E10, f1.2),the corresponding one-dimensional case is an unstable case. However, it not only forms the rect-angular mode in an earlier time but also generates the pockets of unburned gas in a later time.The structures of detonation waves becomes complicated at later time. In the most challengingcase (H50, E50, f1.2), the one-dimensional flow with corresponding parameters is unstable case. Thestructure of detonation wave becomes very complicated when the transverse instabilities dominatethe flow. The vorticity field structures are similar to display a strong two-dimensional incompress-ible flow turbulence [28].
Furthermore, we investigate the relation between the amplitude of perturbation wave and thebehavior of detonation wave. Generally speaking, the larger amplitude releases larger amount ofenergy during the collision with shock front and that generates more small scale structures in thedetonation waves. We also observe that the number of detonation cells in y-direction depends onthe wave number of sinusoidal perturbation.
The paper is organized as follows. In Section 2, a brief introduction to WENO schemes for solv-ing hyperbolic conservation laws is given. In Section 3, a brief introduction to two-dimensionaldetonation model with governing equations and its perturbed initial conditions is given. Differ-ent boundary conditions are briefly introduced and compared quantitatively and qualitatively inSection 3.2. Section 3.3 outlines the general framework of numerical schemes. Three numerical ex-amples are simulated under different resolutions and various perturbation waves and their resultsare discussed in Section 4. Conclusion and remarks are given in Section 5.
2 Weighted Essentially Non-Oscillation Schemes
In this section we briefly describe the general framework of the (2r− 1) order characteristics basedweighted essentially non-oscillatory conservative finite difference scheme (WENO) for solving thesystem of hyperbolic conservation laws in the form
∂u
∂t+∇ · F(u) = 0. (1)
Without loss of generalities, we will present the (2r− 1) order conservative WENO finite difference
4
scheme in one dimension. Explicit formula for fifth (r = 3) order scheme is given. Extension tohigher order (r > 3) WENO scheme is straightforward.
xi xi+1 xi+2xi-1xi-2 xi+1/2
S2
S0
S1
S5 τ5
β0
β2
β1 ω1
ω0
ω2
Figure 1: The computational uniform grid xi and the 5-points stencil S5, composed of three 3-pointssubstencils S0, S1, S2, used for the fifth-order WENO reconstruction step.
Consider a uniform grid defined by the points xi = i∆x, i = 0, . . . , N , which are called cell centers,with cell boundaries given by xi+ 1
2= xi +
∆x2 , where ∆x is the uniform grid spacing (see Fig. 1).
The semi-discretized form of (1) is transformed into the system of ordinary differential equationsby method of line
dui(t)
dt= − ∂f
∂x
∣∣∣∣x=xi
, i = 0, . . . , N, (2)
where ui(t) is a numerical approximation to the cell-averaged value u(xi, t).
To form the flux differences across the uniformly spaced cells, conservative finite-difference formu-lation for hyperbolic conservation laws requires high-order consistent numerical fluxes at the cellboundaries. We implicitly define the numerical flux function h(x) as
f(x) =1
∆x
∫ x+∆x2
x−∆x2
h(ξ)dξ, (3)
such that the spatial derivative in (2) is exactly approximated by a conservative finite differenceformula at the cell boundaries xi± 1
2,
dui(t)
dt=
1
∆x
(hi+ 1
2− hi− 1
2
), (4)
where hi± 12= h(xi± 1
2). High order polynomial interpolations to hi± 1
2are computed using known
cell-averaged values fj = f(xj), j = i− r + 1, . . . , i, . . . , i+ r − 1.
The classical (2r− 1) order WENO scheme uses (2r− 1)-points global stencil, which is subdividedinto r substencils {S0, S1, . . . , Sr−1} with each substencil containing r grid points and a global
5
stencils S2r−1 =
r−1∪i=0
Si. For r = 3, the 5-points global stencil, hereafter named S5, is subdivided
into three 3-points substencils {S0, S1, S2}.
The (2r−1) degree polynomial approximation fi± 12= hi± 1
2+O(∆x2r−1) is built through the convex
combination of the interpolated values fk(xi± 12), in which fk(x) is the r degree polynomial below,
defined in each one of the substencils Sk:
fi± 12=
r−1∑k=0
ωkfk(xi± 1
2), (5)
where
fk(xi+ 12) = fk
i+ 12
=r−1∑j=0
ckjfi−k+j , i = 0, . . . , N. (6)
fki− 1
2
can be found by symmetry. The ckj are Lagrangian interpolation coefficients (see [3]) and ωk
are normalized nonlinear weights (weights), which will be described below.
The regularity of the (r−1) degree interpolation polynomial approximation fk(x) at the substencilSk is measured by the lower order local smoothness indicators βk, which are given by
βk =r−1∑l=1
∆x2l−1
∫ xi+1
2
xi− 1
2
(dl
dxlfk(x)
)2
dx. (7)
For r = 3, the explicit expression of the βk in terms of the cell averaged values of f(x), fi are givenby
β0 =13
12(fi−2 − 2fi−1 + fi)
2 +1
4(fi−2 − 4fi−1 + 3fi)
2 , (8)
β1 =13
12(fi−1 − 2fi + fi+1)
2 +1
4(fi−1 − fi+1)
2 , (9)
β2 =13
12(fi − 2fi+1 + fi+2)
2 +1
4(3fi − 4fi+1 + fi+2)
2 . (10)
In the classical WENO scheme [1, 3] (WENO-JS), the normalized nonlinear weights ωk and un-normalized nonlinear weights αk in each substencil Sk, are defined as
ωk =αk∑r−1l=0 αl
, αk =dk
(βk + ϵ)p, k = 1, . . . , r − 1. (11)
The parameter ϵ (typically 10−12) is used to avoid the division by zero in the denominator andpower parameter p = 2 is chosen to increase the difference of scales of distinct weights at non-smooth parts of the solution. The coefficients {d0, d1, . . . , dr−1} are called the ideal weights since
6
they generate the (2r − 1) order central upwind scheme using the (2r − 1)-points stencil when thesolution is smooth. For r = 3, the ideal weights are
{d0 =
310 , d1 =
35 , d2 =
110
}.
The novel idea of the WENO-Z scheme (WENO-Z) [5, 29] is the modification of the βk with higherorder information obtained from a global optimal order smoothness indicator, which is denotedhere by τ2r−1. This new global optimal order smoothness indicator τ2r−1 is built using cell-averagedvalues in the whole S2r−1 stencil in the form of a linear combination of βk, that is,
τ2r−1 =r−1∑k=0
ckβk, (12)
where ck are given constants (see [5, 29]).
The general definitions of the normalized and un-normalized nonlinear weights ωZk and αZ
k , respec-tively, are
ωZk =
αZk∑r−1
l=0 αZl
, αZk =
dkβZk
= dk
(1 +
(τ2r−1
βk + ϵ
)p), k = 0, . . . , r − 1, (13)
where p ≥ 1 is the power parameter, used to enhance the relative ratio between the smoothnessindicators. For r = 3, the global high order smoothness indicator τ5 is
τ5 = |β0 − β2| . (14)
The numerical results in [5] confirmed the improved performance of the modified smoothness indica-tors over the WENO-JS scheme in computing a higher resolution solution when solving hyperbolicPDEs with discontinuous solutions.
3 Governing Equations
Two-dimensional unsteady reactive Euler equations for a perfect ideal gas coupled with irreversiblechemical reaction in a conservation form is given by
∂Q
∂t+∇ ·
−→F = S, (15)
where−→F = (F,G) is the flux vector and
Q =
ρρuρvEρf1
,F =
ρu
ρu2 + Pρuv
(E + P )uρf1u
,G =
ρvρuv
ρv2 + P(E + P )v
ρf1v
,S =
0000
ω(T, f1)
. (16)
7
where ρ is density or mass density, P is pressure, u and v are the x- and y-component of velocitiesrespectively. f1 is the reactant mass fraction satisfying 0 ≤ f1 ≤ 1, where f1 = 0 describes thecompletely burned state and f1 = 1 describes the unburned state.
The total specific energy E is given by,
E =P
γ − 1+
1
2ρ(u2 + v2) + ρf1q0, (17)
and the source term ω(T, f1) due to the chemical reaction is
ω(T, f1) = −Kρf1e−Ea/T , (18)
where γ is the ratio of specific-heat (assumed to be a constant here), q0 is the heat-release parameter,Ea is the activation-energy parameter, and K is a pre-exponential factor that sets the spatial andtemporal scales. The temperature is defined by
T = P/ρR, (19)
where R is the specific gas constant (with a suitable normalization, R = 1 in this study).
3.1 Initial Conditions
In the case of one-dimensional detonation wave the C-J velocity DCJ and detonation velocity Dcan be determined respectively with the given parameters γ, q0, and the overdrive factor f . Theflow variables ρ, u and P can be computed by specifying a given mass fraction f1 profile. We referto [15, 19] for details.
In this work we specify an exponential function as an initial profile of the mass fraction f1. Thecorresponding one-dimensional typical ZND spatial profiles of different variables are shown in Fig. 2which will be used as the setup of initial conditions of the detonation wave simulations.
In the case of two-dimensional problem, the initial condition in the y-component of velocity is givenby a transversely sinusoidal perturbed planar ZND wave, i.e.,
v(x, y, t = 0) =
0 x < xd −Ww
A sin(2yπkL
)xd −Ww ≤ x ≤ xd
0 x > xd
,
where A, k and L are the amplitude, wave number and wavelength of the sinusoidal perturbationof the detonation wave front and Ww is the width of perturbation zone. xd is the initial locationof the detonation wave front in the x-direction (the detonation waves travel from right to left withthe frame of reference set on the detonation front). Since the problem is periodical in y-direction,L is equal to the height of the domain in y-direction. In this study, Ww = 1 and L = 10 are used.
8
x
Mas
sF
ract
ion
0 1 2 3
0.2
0.4
0.6
0.8
1
x
Den
sity
0 1 2 30
2
4
6
8
10
x
Vel
ocity
0 1 2 3-10
-8
-6
-4
-2
0
x
Pre
ssur
e
0 1 2 30
20
40
60
80
Figure 2: Initial profiles of mass fraction f1, density ρ, velocity u and pressure P for an one-dimensional ZND detonation wave.
3.2 Boundary Conditions
In this study, the problem is assumed to be periodical in the y-direction. Hence, periodical boundarycondition is imposed in the y-direction. Since the frame of reference is set on the detonation frontlocated inside the computational domain, the right boundary condition is set to be the initial freestream inflow condition.
The left boundary condition for the boundary/ghost cells, however, requires a more elaborate dis-cussion as the detonation waves will travel to the left side of the domain if the nonlinear system ofhyperbolic PDEs is integrated long enough in time. Depending on the length of the computationaldomain after truncating the physical domain to a finite size and the numerical boundary conditionsimposed there, undesirable waves might reflect back into the computational domain and pollutethe solution if the reflected wave is sufficiently strong. It should be noted that the left boundaryis not a real physical boundary, like a solid wall, but an artificial one, created by the truncationof the physical domain into a computational domain. Theoretically, the grid points for the bound-ary/ghost points should satisfy the PDEs exactly. However, in order to close the discrete system,a numerical boundary condition must be imposed on the boundary/ghost points.
In this work, we will revisit three kinds of numerical boundary conditions, which have often beenused in the published literature, and demonstrate their effectiveness in handling a truncated physicaldomain in the numerical simulation of detonation waves. An alternative approach based on thePerfectly Match Layer (PML) will then be proposed, described and compared with the other threeapproaches. It should be noted that, in this study, the problem of imposing left boundary conditionfor ghost cells is essentially one-dimensional.
The first approach for imposing numerical boundary condition is to assume the computationaldomain is sufficiently large such that no waves can reach the left truncated boundary at a givenfinal time. In this case, the initial free stream condition can be naturally imposed. It guaranteesthe computed solution is free of pollution of reflected wave from the left computational domainboundary. Its solution can be used as reference solution when comparing different approaches in
9
imposing left numerical boundary condition. The major downside of this approach is the cost ofcomputing if the solution time is very large. Large number of grid points will be needed to maintaina specific grid resolution.
The second approach is to extrapolate the values of the ghost cells from the solution in the interiorof the computational domain. This approach is easy and used often in the literature [30]. However,this approach will result in a well-posed problem only for supersonic flow and one can show thatthe flow at the left computational boundary is subsonic in this study.
The third approach is to employ the characteristic boundary condition for the boundary/ghostpoints by taking into account the direction of the propagating characteristic wave at the boundary.The outgoing characteristic waves will be allowed to move out of the computational domain and theincoming characteristic waves from free stream flow will be allowed to move into the computationaldomain. As discussed in the later Section 3.3, the eigensystem of the Euler equation with specieswith the right eigenvectors R, left eigenvectors R−1 and eigenvalues Λ are provided.
The procedure of characteristic boundary condition can be summarized as follow: (ignoring thedependency in y and assuming the boundary/ghost points are x−r, . . . , x−1 for some r > 0.)
1. Given the conservative variables Q, the linearized eigensystem of the PDEs (R,R−1,Λ) arecomputed at x0, the first interior grid point.
2. The characteristic variable W is formed by projecting Q from the physical space onto thecharacteristic fields via the left eigenvectors as W = R−1Q at all the grid points needed inthe next step.
3. To impose the characteristic boundary condition, the characteristic variable W is modifiedaccording to the sign of the corresponding eigenvalues, that is, with W(x0) = W(x0) andW(x1) = W(x1),
W k(xi−1) =
{2W k(xi)−W k(xi+1) λk(x0) ≥ 0Wk(x = −∞) λk(x0) < 0
, i = 0, . . . ,−r + 1, (20)
where Wk,W k and λk are the k th component of the W, updated characteristic variable Wand Λ, respectively. Here we show the first order extrapolation of outgoing characteristicwaves in a recursive manner.
4. An updated conservative variable Q can then be recovered by projecting W from the char-acteristic fields back onto the physical space via the right eigenvectors as Q = RW.
The fourth approach is inspired from the idea of perfect match layer (PML) for unsteady Eulerequations [27]. In PML, an extended domain (layer) is created to allow the transmission anddamping of waves into the layer in order to minimize the reflection of the waves back into the
10
computational domain. In essence, the conservative variables Q in the PML layer are separatedinto a time-independent steady part Qp and perturbed part Q′ as
Q = Qp +Q′. (21)
Qp is called a time-independent pseudo mean-flow in [27, 31] and satisfies the steady state Eulerequation, that is,
∂Qp
∂t= ∇ · F(Qp) = 0, (22)
where F = (F,G) is the flux vector in two dimensions.
Using (22), one can obtain the equation for Q′, that is,
∂Q′
∂t+∇ · δF = 0, (23)
where δF = F(Q)− F(Qp).
In the frequency domain, it becomes
(−iω)Q′ +∇ · δF = 0, (24)
where the tilde denotes a Laplace transformed quantity.
For clarity, we will present the method for a two dimensional problem. We shall denote δF =F(Q)− F(Qp) and δG = G(Q)−G(Qp).
In the PML layer, by employing the change of complex variables
x −→ x+i
ω
∫ x
xp
σx(ξ)dξ, y −→ y +i
ω
∫ y
yp
σy(ξ)dξ, (25)
where xp (yp) and σx (σy) are the starting locations and smooth damping functions with σx(xp) = 0(σy(yp) = 0) of the PML layer in the x (y)-direction, respectively, one has
(−iω)Q′ +1
1 + iσxω
∂δF
∂x+
1
1 + iσy
ω
∂δG
∂y= 0. (26)
By splitting Q′ into Q′1 and Q′
2 which are the flow variables in x- and y-directions respectively,(26) can be written as
(−iω)Q′1 +
1
1 + iσxω
∂δF
∂x= 0, (27)
(−iω)Q′2 +
1
1 + iσy
ω
∂δG
∂y= 0, (28)
11
where Q′ = Q′1 + Q′
2.
By rearranging the equations above in the time domain, one has
∂Q′1
∂t+ σxQ
′1 +
∂δF
∂x= 0, (29)
∂Q′2
∂t+ σyQ
′2 +
∂δG
∂y= 0. (30)
The two equations above are for absorbing Q′ in the PML layer. By adding them together, anequation for Q′ can be formed, namely,
∂Q′
∂t+ σx(Q
′ −Q′2) +
∂δF
∂x+ σy(Q
′ −Q′1) +
∂δG
∂y= 0. (31)
Since Qp satisfies the steady Euler equation (22), the PML equation for Q can be written as
∂Q
∂t+∇ · F+ σx(Q− Qp −Q′
2) + σy(Q− Qp −Q′1) = 0, (32)
∂Q′2
∂t+ σyQ
′2 +
∂δG
∂y= 0. (33)
In this work, PML is not needed in the periodical y-direction. Therefore, one can neglect (33) andassume σy = 0 and Q′
2 = 0 in the implementation of the above PML in our study. We also assumeQp = Q0 which is the initial condition for steady ZND solution inside the PML layer. We refer to[27] for more details.
To illustrate the efficiency and accuracy of (32), we consider the case (H2, E20, f1.1) with followingparameters (see Section 4.1 for details)
q0 = 2, Ea = 20, f = 1.1,
as proposed in [16, 32] as a classical example of two-dimensional detonation wave. The computation-al domain is (x, y) = [0, 230]× [−5, 5] and the solution is defined as reference solution since no flowstructure has reached the left end of the boundary at x = 0 at the final time. With the implementa-tion of PML boundary condition, the computational domain is set to be (x, y) = [100, 230]× [−5, 5],and PML layer is (xa, xb) = [100, 106] and detonation front is initially located at xd = 225. ThePML layer damping function σx is defined as
σx =
{α∣∣∣ x−xbxa−xb
∣∣∣β xa ≤ x < xb
0 else, (34)
where α = 60 and β = 4.
The comparison of the pressure between the four kinds of boundary condition and the referencesolution is shown in Fig. 3. Reflection of waves from the left boundary are observed in every kinds of
12
boundary condition discussed above. The PML boundary condition has the minimal pressure wavereflection from the left computational boundary with maximum absolute error less than 2 × 10−5
while the error of other three kinds of boundary condition can be up to 2 × 10−2. Hence, we willemploy the PML boundary condition in this study of detonation waves.
Characteristics Extrapolation
100 105 110 115 120
1.0E-07 3.5E-04 8.5E-04
100 105 110 115 120
1.0E-07 3.5E-04 8.5E-04
Free stream PML
100 105 110 115 120
1.0E-07 3.5E-04 8.5E-04
100 105 110 115 120
1.0E-07 3.5E-04 8.5E-04
Figure 3: (H2, E20, f1.1): Absolute error in pressure of the characteristic, zero-order extrapolation,free stream and PML boundary conditions when compared with the reference solution computedfrom a large computational domain.
3.3 Numerical Methods
We approximate the inviscid flux, via the well-known fifth order characteristic-based weightedessentially non-oscillatory conservative finite difference scheme (WENO). Following [1, 3, 5], thehyperbolicity of the Euler equations admits a complete set of right and left eigenvectors for theJacobian of the system.
By defining the two dimensional unit outward normal vector n = (nx, ny) = (cos θ, sin θ), whereθ = tan−1(ny/nx), and the rotation matrix
T =
1 0 0 0 00 nx ny 0 00 −ny nx 0 00 0 0 1 00 0 0 0 1
, T−1 =
1 0 0 0 00 nx −ny 0 00 ny nx 0 00 0 0 1 00 0 0 0 1
, (35)
13
the Euler system has Jacobian in general form as
A = (FJ ,GJ) · n, (36)
where FJ and GJ are Jacobian matrices of fluxes of F and G, respectively. One can then find theright eigenvectors R(Q), left eigenvectors R−1(Q) and eigenvalues Λ(Q) of FJ . In the Cartesianframe, the eigensystem is a function of Q.
By the rotational invariance property of Euler equations [33], the eigensystem of equation (16) canbe represented generally as
A = RΛR−1, (37)
where Q = TQ, R = T−1R(Q), R−1 = R−1(Q)T, Λ = Λ(Q). In a rotational frame, the eigensys-tem is a function of Q.
Hence, the Jacobian matrix of system (16) becomes,
A =
0 nx
12(γ − 3)u2nx − uvny (3− γ)unx + vny
12(γ − 3)v2ny − uvnx (1− γ)uny + vnx
((γ − 1)u2 − h)u · n (h− u2(γ − 1))nx + (1− γuv)ny)
−f1u · n f1nx
ny 0 0
(1− γ)vnx + uny (γ − 1)nx (1− γ)q0nx
(3− γ)vny + unx (γ − 1)ny (1− γ)q0ny
(h− v2(γ − 1))ny + (1− γuv)nx γu · n (1− γ)q0u · n
f1ny 0 u · n
. (38)
The right and left eigenvector matrices of A are
R =
1 1 1 0 1
u+ c nx u− c nx (u · n) nx −ny (u · n) nx
v + c ny v − c ny (u · n) ny nx (u · n) ny
h+ c u · n h− c u · n ζ2 u · k 0
f1 f1 0 0 − ζ2q0
, (39)
14
R−1 = c
u2
2 − c u·nγ−1
cγ−1nx − u c
γ−1ny − v
u2
2 + c u·nγ−1 − c
γ−1nx − u − cγ−1ny − v
2(h− u2)− 4f1q0(u·n)2ζ 2u+ 4f1q0u
ζ 2v + 4f1q0vζ
−(u · k) u2 2u(u · k)− ny
c 2v(u · k) + nxc
2fq0u2
ζ −4fq0uζ −4fq0v
ζ
1 −q0
1 −q0
−2− 4f1q0ζ
4q0(h−(u·k)2)ζ
−2u · k 2q0u · k4f1q0
ζ2q0(u2−2h)
ζ
, (40)
and their corresponding eigenvalues are
λ1 = u · n+ c, λ2 = u · n− c, λ3 = λ4 = λ5 = u · n. (41)
Here, h = (E+P )/ρ and c =√γP/ρ are the specific enthalpy and sound speed, respectively. Also,
we define c = (γ− 1)/(2c2),u ·n = unx+ vny,u ·k = −uny + vnx,u2 = u ·u = (u ·n)2+(u ·k)2 =
u2 + v2, ζ = (u · n)2 − (u · k)2.
The approximated eigenvalues and eigenvectors are obtained via the Roe linearized Riemann solver.The first order global Lax-Friedrichs flux is used as the low order building block for the high orderreconstruction step of the WENO scheme. After projecting the positive and negative fluxes on thecharacteristic fields via the left eigenvectors, the high order WENO reconstruction step is appliedto obtain the high order approximation at the cell boundaries using the surrounding cell-centeredvalues, which are then projected back into the physical space via the right eigenvectors and addedtogether to form a high order numerical flux at the cell-interfaces. The conservative difference ofthe reconstructed high order fluxes can then be computed for inviscid flux.
The resulting system of ordinary differentiation equations ODE (4) is advanced in time via the thirdorder TVD Runge-Kutta scheme. The CFL condition is set to be CFL = 0.45 in the numericalexperiments performed in this study. We refer to [3, 5] for further details on the WENO algorithmfor solving the hyperbolic conservation laws.
15
4 Numerical Experiments and Discussions
In this section, the fifth order WENO-Z scheme is employed to simulate three classical two-dimensional examples of detonation waves, all of which have been studied previously, albeit usingdifferent techniques.
4.1 Small heat release and low activation energy
In the case (H2, E20, f1.1), parameters are chosen in Table I whose corresponding stiffness coefficientand detonation front speed are K = 1134363.64 and D = 2.04 respectively. The one-dimensionalflow with corresponding parameters is a linearly stable case because there is no linearly unstablelongitudinal modes [34, 35]. The parameters of perturbation wave are (k = 4, A = 0.1). Thecomputational domain is set to be (x, y) = [100, 230]×[−5, 5], and PML layer is (x, y) = [100, 106]×[−5, 5] and detonation front is initially located at xd = 225.
Fig. 4 (a) and (b) display mesh refinement study of WENO-Z scheme, peak pressure and detonationfront speed are measured in reaction zone at time t = 100. The computed solution converges tosteady state solution as grid resolution increases, and it agrees well with [36] and our previous work[19]. The temporal evolution of the peak pressure Pm(t) in reaction zone under grid resolutions δ5,δ10 and δ20 are shown in Fig. 4 (c). In low grid resolution δ5, the scheme does not reach the steadystate solution. As high grid resolutions δ10 and δ20, both results converge to steady state solutionfor time t > 30.
(a) Peak pressure Pm(t) (b) CJ Speed DCJ (c) Peak Pressure Pm(t)
Relative Mesh Size
Pea
k P
ress
ure
0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.113.6
3.62
3.64
3.66
3.68
3.7
3.72
3.74
WENO-Z
Steady State Solution
Relative Mesh Size
Det
onat
ion
Fro
nt S
peed
0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.111.95
2
2.05
2.1
2.15
2.2
2.25
2.3
2.35
2.4 WENO-Z
Steady State Solution
Time
Pea
k P
ress
ure
0 20 40 60 80 100
3.5
4
4.5
delta=5delta=10delta=20
Figure 4: (H2, E20, f1.1): (Color online) Mesh study of (a) peak pressure Pm(t) and (b) detonationfront speed DCJ at time t = 100. (c) Pm(t) with grid resolutions δ5, δ10 and δ20.
Pressure and vorticity contour with grid resolutions δ10 and δ20 at time t = 30 are given in theFig. 5. The flow variables generated regular cellular structures which are due to the large amount
16
of energy being released during the collisions between the perturbation waves and shock fronts.They are in very good agreement with the results presented in the literature [16, 32]. Furthermore,the results of high grid resolution simulations capture more small scale structures than the low gridresolution one in the Fig. 5.
Pressure Vorticity
x
y
195 200 205 210-4
-2
0
2
4 2.46 2.54 2.62 2.7 2.78 2.86 2.94
x
y
195 200 205 210-4
-2
0
2
4
x
y
195 200 205 210-4
-2
0
2
4 -0.26 -0.18 -0.1 -0.02 0.06 0.14 0.22
x
y
195 200 205 210-4
-2
0
2
4
Figure 5: (H2, E20, f1.1): (Color online) The profiles of pressure and vorticity at time t = 30 withgrid resolutions δ10 (first row) and δ20 (second row).
To explore the relation between amplitude of perturbation wave A and structures of the detonationwave, we increase the amplitude from A = 0.1 to A = 0.5 and show the temporal evolutionof the peak pressure Pm(t) at time t = 100 and pressure at time t = 10 with grid resolutionδ20. In the Fig. 6 (a), initially, large amplitude case generates high peak pressure in reactionzone as expected because more energy is being released when the shock fronts interacted with thestronger perturbation wave. Because of the stability in this case, perturbations decay with timethat indicates the steady solution is hydrodynamically stable and probably unique [11]. In Fig. 6(b), the cellular structures are similar for both cases, and results of large amplitude are betterdeveloped than those of small amplitude. In Fig. 6 (c) and (d), we show the pressure structure ofdetonation front for large amplitude A = 0.5 and grid resolution δ20 before and after triple pointscollision. The pressure structure of detonation front agrees well with [11, 32].
4.2 Large heat release and low activation energy
The case (H50, E10, f1.2) has no prediction of regular cell structures from the theory of instabilitybased on geometric acoustics [32] and the corresponding one-dimensional flow is unstable. The
17
(a) Pm(t) (b) Pressure
Time
Pea
k P
ress
ure
0 20 40 60 80 100
3.5
4
4.5
5
5.5 WENO-Z, A=0.1WENO-Z, A=0.5
x
y
205 210 215 220-4
-2
0
2
4 2.46 2.54 2.62 2.7 2.78 2.86 2.94 3.02
x
y
205 210 215 220-4
-2
0
2
4
(c) Pressure before collision (d) Pressure after collision
x
y
223 224 2250.5
1
1.5
2
2.5
3
3.5
4
4.5
Level P
31 4.1
25 3.5
19 2.9
13 2.3
7 1.7
1 1.1
TRIPLE POINT
MACH STEM
DETONATION
x
y
223 224 2250.5
1
1.5
2
2.5
3
3.5
4
4.5
Level P
31 4.1
25 3.5
19 2.9
13 2.3
7 1.7
1 1.1
TRIPLE POINT
MACH STEM
DETONATION
Figure 6: (H2, E20, f1.1): (Color online) (a) The peak pressure Pm(t) for perturbation amplitudesA = 0.1 and A = 0.5. (b) Pressure contour of A = 0.1 (upper) and A = 0.5 (lower). (c) Pressurecontour before the triple points collision. (d) Pressure contour after the triple points collision. Thegrid resolution is δ20.
18
stiffness coefficient and detonation front speed are K = 3.124 and D = 2.04 respectively. Thecomputational domain is (x, y) = [300, 410]× [−5, 5] and PML layer is (x, y) = [300, 306]× [−5, 5].The ZND wave is initially located at xd = 400.
In Fig. 7, we show the comparative results of the temporal evolution of peak pressure Pm(t) attime t = 50 with different grid resolutions δ5, δ10, δ20 and various perturbation waves (k = 2, A =0.5), (k = 2, A = 1.0), (k = 3, A = 0.5). The grid resolution study of Pm(t) shows that WENO-Zscheme with lower grid resolution δ5 cannot capture the small scale structures and the naturalphenomena of detonation wave. In contrast, higher peak pressures Pm(t) with regular pattern arecaptured with high grid resolution δ10, δ20 at late time. The peak pressure also increases to a highervalue when we perturbed the initial condition with larger amplitude A = 1.0. Furthermore, werecord the temporal evolution of the pressure P and velocity u at location (x, y) = (355, 0) whichare shown in Fig. 8.
Time
Pea
k P
ress
ure
0 20 40
60
80
100
120 delta=5delta=10delta=20
Time
Pea
k P
ress
ure
0 10 20 30 40 50
50
60
70
80
90
100
110
120
130
k=2, A=1.0k=2, A=0.5k=3, A=0.5
Figure 7: (H50, E10, f1.2): (Color online) The temporal evolution of peak pressure Pm(t) with(Left) δ5, δ10, δ20 under the perturbation wave parameters (k = 2, A = 0.5) and (Right) three setsof perturbation wave parameters (k,A). The grid resolution is δ20.
The contours of flow variables computed by WENO-Z scheme at time t = 10 with grid resolutionδ20 are shown in Fig. 9 whose structures agree well with those in the literature [16, 32]. Weobserve that two and three detonation cells in y-direction are formed in the case of k = 2 andk = 3 respectively. It agrees well with the conclusion that the final number of cells depends on thespecific initial perturbation. We refer to [32] for details.
The detonation structures in the flow variables become more complicated than those generated inthe previous case of small heat release and low activation energy, such as, the regular rectangle
19
Time
Pre
ssu
re
0 10 20 30 40 5034
35
36
37
38k=2, A=1.0k=2, A=0.5k=3, A=0.5
TimeV
elo
city
0 10 20 30 40 50-3.6
-3.4
-3.2
-3
-2.8
-2.6
-2.4
-2.2
-2
k=2, A=1.0k=2, A=0.5k=3, A=0.5
Figure 8: (H50, E10, f1.2): (Color online) (Left) Pressure P and (Right) Velocity u with grid reso-lution δ20 at locations (x, y) = (355, 0).
structure in the pressure disappear. Furthermore, the vortex sheet are well developed into regularpairs from two colliding triple points when they detached from front. The mushroom shapedvortex rollup structures are formed in vorticity, temperature and mass fraction. The structuresin the results of larger amplitude A = 1.0 are more complicated than those of smaller amplitudeA = 0.5, such as, the vortex rolls up earlier with larger amplitude A = 1.0 than the smalleramplitude A = 0.5 (k = 2). We also observe that, the vortex starts rolling up further upstream atlocation x = 375 with k = 2 but not with k = 3 (A = 0.5).
4.3 Large heat release and large activation energy
In this section we present numerical results of a challenging case (H50, E50, f1.2), which is theunstable ZND wave at arbitrarily short wavelengths [16, 32]. The corresponding stiffness coefficientand detonation front speed of this system is K = 871.42 and D = 7.4594 respectively. Thecomputational domain is (x, y) = [300, 410]× [−5, 5] and PML layer is (x, y) = [300, 306]× [−5, 5].The ZND wave is initially located at xd = 400.
The comparative profiles of pressure, temperature, vorticity and mass fraction between two per-turbation cases (k = 2, A = 0.1) and (k = 2, A = 0.2) computed by WENO-Z scheme with gridresolution δ20 at times t = 10, 15, 20 are present in the Fig. 10 and Fig. 11. The result agreeswell with those in the literature [16, 32]. The structures of detonation wave becomes increasingly
20
k = 2, A = 1.0 k = 2, A = 0.5 k = 3, A = 0.5
Pressure
Tem
perature
Vorticity
Mass
Fraction
Figure 9: (H50, E10, f1.2): (Color online) Contours of the pressure, temperature, vorticity andmass fraction with grid resolution δ20 under the perturbation waves with parameters (k = 2, A =1.0), (k = 2, A = 0.5) and (k = 3, A = 0.5).
21
chaotic with time increases. At the early time t = 10, two regular detonation cellular structures areformed and the vortices are well developed and symmetrically distributed in the wake of the front.The detonation wave structure starts becoming irregular at time t = 15 due to the material burnsfast resulting in high over-pressures. Meanwhile, strong pockets of unburned gas are constantlygenerated and subsequently burnt. The detonation wave structure becomes chaotic at time t = 20when the transverse instabilities grow strong and dominate the flow. Furthermore, in both cases,the vorticity generates similar two-dimensional incompressible turbulence at t = 20 as reported in[28]. In the case of large amplitude (k = 2, A = 0.2), more small scale structures in the detonationswave are generated due to larger amount of energy being released during the collision with shockfront more than those in the case of small amplitude (k = 2, A = 0.1).
t = 10 t = 15 t = 20
Pressure
Tem
perature
Vorticity
Mass
Fraction
Figure 10: (H50, E50, f1.2): (Color online) Contours of the pressure, temperature, vorticity and massfraction at times t = 10, 15, 20 with grid resolution δ20. The parameters used for the perturbationwaves are (k = 2, A = 0.1).
We choose two symmetric locations, namely (185, 2.5) and (185, − 2.5), to record the evolution ofpressure P and velocity u in the perturbation cases (k = 2, A = 0.1) and (k = 2, A = 0.2) as shownin Fig. 12. At the early time, their evolution are identical, the symmetry property is gradually lostdue to the irregularity of detonation as time increases. And larger amplitude (k = 2, A = 0.2) caseloses the flow symmetry earlier than smaller amplitude (k = 2, A = 0.1) case, which conforms tothe theory that strong turbulence flow contributes the irregularity of cellar pattern early.
22
t = 10 t = 15 t = 20
Pressure
Tem
perature
Vorticity
Mass
Fraction
Figure 11: (H50, E50, f1.2): (Color online) Contours of the pressure, temperature, vorticity and massfraction at times t = 10, 15, 20 with grid resolution δ20. The parameters used for the perturbationwaves are (k = 2, A = 0.2).
(A = 0.1) (A = 0.2)Pressure Velocity Pressure Velocity
Time
Pre
ssur
e
0 10 20
10
20
30
40
50
60
70
80[185,2.5], A=0.1[185,-2.5], A=0.1
Time
Vel
ocity
0 10 20 30
-5
-4
-3
-2
-1
0[185,2.5], A=0.1[185,-2.5], A=0.1
Time
Pre
ssur
e
0 10 20
10
20
30
40
50
60
70
80[185,2.5],A=0.2[185,-2.5], A=0.2
Time
Vel
ocity
0 10 20 30
-5
-4
-3
-2
-1
0[185,2.5],A=0.2[185,-2.5], A=0.2
Figure 12: (H50, E50, f1.2): (Color online) Pressures and velocities with the perturbation waveswith parameters (k = 2, A = 0.1), (k = 2, A = 0.2). The data are recorded at location (185, 2.5)and (185, − 2.5).
23
The pressure, temperature, vorticity and mass fraction of the flow computed by WENO-Z schemewith grid resolution δ20 and (k = 3, A = 0.1) are shown in the Fig. 13. One can see that thenumber of detonation cells depends on the wave number of the perturbation sinusoidal waves.
t = 5 t = 10 t = 15
Pressure
Tem
perature
Vorticity
Mass
Fraction
Figure 13: (H50, E50, f1.2): (Color online) Contours of the pressure, temperature, vorticity andmass fraction at times t = 5, 10, 15 with grid resolution δ20. The parameters of the perturbationwaves are (k = 3, A = 0.1).
5 Conclusion and Future Work
In this work, we studied the performance of the improved fifth order characteristic based weightedessentially non-oscillatory (WENO-Z) finite difference scheme for two dimensional detonation wavesimulations. We presented some preliminary numerical studies of three examples with small heatrelease and low activation energy, large heat release and low activation energy, large heat release andlarge activation energy, respectively. The governing equations are a system of nonlinear hyperbolicconservation laws with a species production source term based on a simple irreversible chemicalreaction. The heat generated by the chemical reaction is added to the total energy of the system,which drives the detonation front and vice versa. The behavior of the combustion front and thereaction zone behind the front depends exponentially on the local mass fraction and the temperaturearound the combustion front. The fifth order WENO-Z scheme with different grid resolutions and
24
third order TVD Runge-Kutta scheme were employed to solve the nonlinear hyperbolic system inorder to capture sharp detonation fronts and to resolve small scales structures that appear in thehighly complex solution of flow. The computer program is written based on the FORTRAN 95subroutines contained in the high performance software library WENOpack [37].
To reduce the computational cost and to avoid wave reflection from the artificial computation-al boundary of a truncated physical domain, we derive an efficient and easily implemented onedimensional Perfectly Matched Layer (PML) absorbing boundary condition (ABC) for the two di-mensional unsteady reactive Euler equation when one of the direction of domain is periodical andinflow/outflow in the other direction. Numerical comparisons among characteristic, free stream,extrapolation and PML boundary conditions are conducted for the detonation wave simulations.It showed that PML boundary condition is the best choice to absorb the outgoing waves whileminimizing the wave reflection from the artificial computational boundary.
Numerical schemes for solving the system of hyperbolic conversation laws which uses a singe modesinusoidal perturbed ZND analytical solution as initial conditions are presented. We showed thatthe larger amplitude of perturbation wave generated more smaller scale structures of detonationwave due to more energy is released when the shock fronts came across the strong perturbation wave.The number of cell structures depends on the wave number of sinusoidal perturbation. Regularrectangular combustion cell, pockets of unburned gas and bubbles and spikes are generated andgenerally well resolved in the simulations. The present results from the simulations agree well withthe published theoretical and numerical results in the literature.
In our future work in this area, a multi-modes perturbation and/or random perturbation within thereaction zone will be considered in order to study the complex detonation behavior. We will examinequantitative behavior of the system via the mixing profile, statistics and spectra of perturbationenergy fields (for example, see [38]) in our future work.
Acknowledgments
The authors would like to acknowledge the funding support of this research by the RGC grantHKBU-200910 from Hong Kong University Grant Council. The first author gratefully acknowl-edges the invitation of Prof. Don, Wai-Sun and also extends his gratitude to the Department ofMathematics at Hong Kong Baptist University for hosting his visit. He also acknowledges Prof.Cheng Wang for many useful discussions throughout the research and the funding support of thisresearch by the Fundamental Research Funds for the Central Universities (1501-841111012).
25
References
[1] D. Balsara and C. W. Shu, Monotonicity preserving weighted essentially non-oscillatoryschemes with increasingly high order of accuracy, J. Comput. Phys. 160 (2000), pp. 405–452.
[2] A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49(1983), pp. 357–393.
[3] G. S. Jiang and C. W. Shu, Efficient Implementation of Weighted ENO Schemes, J. Comput.Phys. 126 (1996), pp. 202–228.
[4] A. K. Henrick, T. D. Aslam and J. M. Powers, Mapped weighted essentially non-oscillatoryschemes: Achieving optimal order near critical points, J. Comput. Phys. 207 (2005), pp. 542–567.
[5] R. Borges, M. Carmona, B. Costa and W. S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys. 227 (2008), pp. 3101–3211.
[6] Y. B. Zeldovich, On the theory of the propagation of detonation in gaseous system, Zh. Eksp.Teor. Fiz. 10 (1940), pp. 542–568.
[7] J. V. Neumann, Theory of Detonation Waves, John von Neumann, Collected Works, Macmil-lan, New York, 6 (1942).
[8] W. Doering, On detonation processes in gases, Ann. Phys. 43 (1943), pp. 421–436.
[9] J. J. Erpenbeck, Stability of steady-state equilibrium detonations, Phys. Fluids 5 (1962), pp.604–614.
[10] A. K. Oppenheim and R. I. Soloukhin, Experiments in gasdynamics of explosions, Ann. Rev.Fluid Mech. 5 (1973), pp. 31–58.
[11] W. Fickett and W. C. Davis, Detonation, University of California Press, Berkeley (1979).
[12] J. H. S. Lee and I. O. Moen, The mechanism of transition from deflagration to detonation invapor cloud explosion, Progr. Energy Combust. Sci. 6 (1980), pp. 359–389.
[13] F. Sharpe and S. A. E. G. Sharpe, Two-dimensional numerical simulations of idealized deto-nations, Proc. R. Soc. Lond. A 456 (2000), pp. 2081–2100.
[14] B. N. Azarenok and T. Tang, Second-order Godunov-type scheme for reactive flow calculationson moving meshes, J. Comput. Phys. 206 (2005), pp. 48–80.
[15] Z. C. Zhang, S. T. Yu, H. He and S. C. Chang, Direct calculation of two- and three- dimensionaldetonations by an extended CE/SE method, AIAA Paper 2001–0476 (2001).
26
[16] M. V. Papalexandris, A. Leonard and P. E. Dimotakis, Unsplit Algorithms for Multidimen-sional System of Hyperbolic Conservation Laws with Source Terms, Comput. Math. Appl. 44(2002), pp. 25–49.
[17] X. He and A. R. Karagozian Numerical Simulation of Pulse Detonation Engine Phenomena,J. Sci. Comput. 19 (2003), pp. 201–224.
[18] C. Wang, J. Ning and J. Lei, Numerical Study on Propagation of Explosion Wave in H2-O2Mixtures, Adv. Nat. Comput. 4222 (2006), pp. 816–819.
[19] Z. Gao, Z. Li and W. S. Don, High Order Weighted Essentially Non-OscillationSchemes for One-Dimensional Detonation Wave Simulations, J. Comput. Math.,(http://www.math.hkbu.edu.hk/ICM/researchreports 11-02)
[20] A. Bourlioux, A. J. Majda and V. Roytburd, Theoretical and Numerical Structure for UnstableOne-Dimensional Detonations, SIAM J. Appl. Math. 51 (1991), pp. 303–343.
[21] M. V. Papalexandris, A. Leonard and P. E. Dimotakis, Unsplit schemes for hyperbolic con-servation laws with source terms in one space detonation, J. Comput. Phys. 134 (1997), pp.31–61.
[22] J. J. Quirk, Godunov-type schemes applied to detonation flows, ICASE Report 93–15 (1993).
[23] J. P. Berenger, A Perfectly Matched Layer for the Absorption of Electromagnetic waves, J.Comput. Phys. 114 (1994), pp. 185–200.
[24] J. S. Hesthaven, On the analysis and construction of perfectly matched layers for the linearizedEuler equations, J. Comput. Phys. 142 (1998), pp. 129–147.
[25] T. Hagstrom, A new construction of perfectly matched layers for hyperbolic systems with appli-cations to the linearized Euler equations, in: G. Cohen, E. Heikkola, P. Joly, P. Neittaanmaki(Eds.), Mathematical and Numerical Aspects of Wave Propagation Phenomena, Springer-Verlag, Berlin (2003), pp. 125–129.
[26] F. Q. Hu, A perfectly matched layer absorbing boundary condition for linearized Euler equationswith a non-uniform mean-flow, J. Comput. Phys. 208 (2005), pp. 469–492.
[27] F. Q. Hu, On the construction of PML absorbing boundary condition for the non-linear Eulerequations, AIAA Paper 2006-0798 (2006).
[28] M. E. Brachet, M. Meneguzzi and P. L. Sulem, Small-Scale Dynamics of High-Reynolds-Number Two-Dimensional Turbulence, Phys. Rev. Lett. 57 (1986), pp. 683–686.
[29] M. Castro, B. Costa and W. S. Don, High order weighted essentially non-oscillatory WENO-Zschemes for hyperbolic conservation laws, J. Comput. Phys. 230 (2011), pp. 1766–1792.
27
[30] H. S. Dou, H. M. Tsai, B. C. Khoo and J. Qiu, Simulations of Detonation Wave Propagation inRectangular Ducts Using a Three-Dimensional WENO Scheme, Combust. Flame, 154 (2008),pp. 644–659.
[31] F. Q. Hu On Perfectly Matched Layer as an absorbing boundary condition, AIAA paper 96–1664 (1996).
[32] A. Bourlioux and A. J. Majda, Theoretical and Numerical Structure for Unstable Two-Dimensional Detonations, Combust. Flame 90 (1992), pp. 211–229.
[33] E. F. Toro, Riemann solvers and numerical methods for fluid dynamics, A practical introduc-tion, Springer, 1999.
[34] J. J. Erpenbeck, Stability of idealized one-reaction detonations, Phys. Fluids 9 (1964), pp.684–696.
[35] H. I. Lee and D. S. Stewart, Calculation of Linear Detonation Instability, One-DimensionalInstability of Plane Detonation, J. Fluid Mech. 216 (1990), pp. 103–132.
[36] H. S. Dou, H. M. Tsai, B. C. Khoo and J. Qiu, Simulations of detonation wave propagation inrectangular ducts using a three-dimensional WENO scheme, Combust. Flame 154 (2008), pp.644–659.
[37] W. S. Don, WENOpack, http://www.math.hkbu.edu.hk/∼wsdon/.
[38] M. Latini, O. Schilling and W. S. Don, Effects of order of WENO flux reconstruction andspatial resolution on reshocked two-dimensional Richtmyer-Meshkov instability, J. Comput.Phys. 221 (2007), pp. 805–836.
28
